Parametric Equation of a Hyperbola

I. Introduction to Parametric Equations:

  • Parametric equations express the coordinates of a point in terms of one or more parameters.
  • In the context of conic sections, parametric equations are particularly useful for describing the motion of a point along a curve.

II. Parametric Equation of a Hyperbola: For a hyperbola, the parametric equations are typically expressed in terms of trigonometric functions. The standard parametric equations for a hyperbola centered at the origin are:

x(t)=asec(t) y(t)=btan(t)

  • (x(t),y(t)) represents a point on the hyperbola.
  • t is the parameter that varies, and it usually ranges over the interval (π2,π2) or (0,2π).

III. Key Parameters:

  • a: Semi-major axis, distance from the center to the vertices along the x-axis.
  • b: Semi-minor axis, distance from the center to the vertices along the y-axis.

IV. Graphical Interpretation:

  • As the parameter t varies, the parametric equations generate points on the hyperbola.
  • The parametric equations describe the relationship between x and y in a dynamic way, capturing the entire shape of the hyperbola as t varies.

Example : 

Suppose we have the parametric equations:

x(t)=3sec(t)

y(t)=2tan(t)

1. Identify Key Parameters:

  • Semi-Major Axis (a): a=3 (coefficient of sec(t))
  • Semi-Minor Axis (b): b=2 (coefficient of tan(t))

2. Eccentricity (e):

  • Use the formula e=1+b2a2 to find eccentricity.
  • e=1+2232=139

3. Foci ((F1,F2)):

  • Use the formula c=a2+b2  to find the distance from the center to each focus.
  • c=32+22=13
  • Foci are located at (±13,0).